Linearisations of triangulated categories with respect to finite group actions

Given an action of a finite group on a triangulated category, we investigate under which conditions one can construct a linearised triangulated category using DG-enhancements. In particular, if the group is a finite group of automorphisms of a smooth projective variety and the category is the bounded derived category of coherent sheaves, then our construction produces the bounded derived category of coherent sheaves on the smooth quotient variety resp. stack. We also consider the action given by the tensor product with a torsion canonical bundle and the action of a finite group on the category generated by a spherical object.

💡 Research Summary

The paper addresses the problem of constructing a “linearised” triangulated category when a finite group acts on a given triangulated category. The author observes that, while linearisation is a standard tool for sheaves (e.g. the category of G‑linearised coherent sheaves on a variety X is equivalent to Coh(